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Stochastic differential equation

A stochastic differential equation (SDE) of Ito-type is written as,
\begin{eqnarray*} dX(t)&=&\mu (t,X(t))dt+\sigma(t,X(t))dW(t),\\ X(0)&=&x_0. \end{eqnarray*} The term $\mu(t,X(t))$ is called drift coefficient and the term $\sigma;(t,X(t))$ is called diffusion coefficient and W(t) is a standard Brownian motion. An SDE is really a stochastic integral equation \[ X(t)=x_0+\int_0^t\mu(s,X(s))ds+\int_0^t\sigma(s,X(s))dW(s).\] The first integral $\int_0^t\mu(s,X(s))ds$ is just an ordinary Riemann Integral whereas the second integral $\int_0^t\sigma(s,X(s))dW(s)$ is a stochastic integral or Ito integral.
If the drift and diffusion coefficient satisfies
  1. $\mu(t,X(t))$ and $\sigma(t,X(t))$ are adapted to the filtration generated by the Brownian motion W.
  2. $|\mu(t,x)-\mu(t,y)|\leq K|x-y|$, for $x,y\in\mathbb{R}^d, t\geq 0$
    $|\sigma;(t,x)-\sigma;(t,y)|\leq K|x-y|$, for $x,y\in\mathbb{R}^d, t\geq 0$
    (Lipschitz-condition),
  3. $|\mu(t,x)|^2+|\sigma(t,x)|^2\leq C(1+|x|^2)$, for $x\in\mathbb{R}^d, t\geq 0$
    (Linear growth bound),
then there exist a unique global strong continuous solution to the SDE. The Lipschitz-condition gives uniqueness and linear growth bound gives global existence. If the linear growth bound fails then the solution might explode in finite time. Take e.g. the SDE, $dX(t)=X(t)^3dt-X(t)^2dW(t),X(0)=1.$
This SDE has the solution $X(t)=1/(1+W(t))$ which explodes when W(t) hits -1 for the first time. This explosion time has the same distribution as $1/G^2$ where G is a standard Gaussian random variable. This means that approximately $3/4$ of the trajectories will explode within 10 time units. [] An SDE which is important in financial applications is the linear SDE $dX(t)=\mu X(t)dt+\sigma X(t)dW(t),X(0)=x_0$. The solution is called a Geometric Brownian motion.

Another important related equation $dX(t)=\kappa(\theta-X(t))dt+\sigma dW(t)$, gives rise to the Ornstein-Uhlenbeck process.

 

Questions: Magnus Wiktorsson
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